A union bound and Bernstein's inequality, see Corollary 2.11 in [1]}, yield that for any \(A > 0\) ,
\(\begin{split}&\mathbb {P}\left( \max _{k = 1,\ldots ,K_n} \delta _n \max _{\lceil C \delta _n \rceil \le j \le n} \left(\frac{\sum _{i = 1}^{j} Y_{nki}}{j}\right)^2 > A^2 (\log K_n)^2 \right)\\\le & \sum _{k = 1}^{K_n} \sum _{j = \lceil C\delta _n \rceil ,\ldots ,n} \left\lbrace \mathbb {P}\left( \sum _{i = 1}^{j} Y_{nki} > \frac{j}{\sqrt{\delta _n}} A \log K_n\right) + \mathbb {P}\left( \sum _{i = 1}^{j} Y_{nki} < -\frac{j}{\sqrt{\delta _n}} A \log K_n\right)\right\rbrace \\\le & 2K_n \sum _{j = \lceil C\delta _n \rceil ,\ldots ,n} \exp \left(-\frac{j^2 A^2(\log K_n)^2/ \delta _n}{2\big (j + c A \log (K_n) j / \sqrt{\delta _n}\big )}\right).\end{split}\) 
